# Subproject A1: Multi Loop Calculations and Computer Algebraic Techniques in Quantum Field Theory

Project
leader:

Prof. Dr. Johann Kühn, Institute of Theoretical
Particle Physics, Karlsruhe Institute of Technology (KIT)

**Summary**

**
Short Summary:**

Project A1 aims at developing concepts and computer algebra programs to evaluate multi-loop amplitudes and at applying the results to particle phenomenology. Methods and results are mainly directed towards QCD, some of them, however, are also of interest for electroweak interactions. Finally, predictions for certain anomalous dimensions as obtained with integrability methods and AdS/CFT correspondence will be checked with these methods in a completely different perturbative framework.

** Summary:
**

Apart from using results and techniques obtained and developed during
the last funding period for numerous phenomenological applications we
will specifically exploit our worldwide unique expertise, methods and
programs, which allow the evaluation of *
massless four-loop propagator diagrams* including their the finite parts.
In this respect the first
funding period can be considered as exploratory phase, where important
conceptual developments had been launched and first results had been
obtained, demonstrating the feasibility of our approach and leading to
important implications, e.g. for the determination of the strange
quark mass.

During the second funding period a number of results have been
published which from the start of the project were considered to be
among the main goals and of utmost theoretical and phenomenological
importance. This refers to the Adler function and the *R*-ratio,
closely related to the cross section for hadron production in
electron-positron annihilation, a result crucial for
reducing the theoretical error in the evaluation of the strong coupling
constant. Using similar techniques, the α^{4}_{s} correction to
the Bjorken sum rule was derived and the generalized Crewther
relation, connecting sum rule and Adler function, verified in this
same order. The evaluation of the three-loop quark and gluon form
factors was based on a closely related strategy.

In the beginning of the third funding period these techniques will be
applied to calculate a variety of anomalous dimensions and
coefficient functions relevant for deep inelastic scattering, again in
order α^{4}_{s}. Using a combination of multi-loop techniques for
massless propagator integrals and massive vacuum integrals described
below, we plan to attack the calculation of a variety of anomalous dimensions
in five-loop order which would be a world record. For *N*=4
supersymmetric gauge theory this technique will help to test important
conjectures based on AdS/CFT correspondence and integrability.

The evaluation of massive vacuum integrals (``tadpole integrals'') in four-loop approximation has started during the first funding period with first results emerging about four years ago. In the meantime it has been possible to calculate two additional derivatives of vector, axial, scalar and pseudoscalar correlators in analytical and ten of them in numerical form, leading to the world most precise determination of the charm and bottom quark masses. In the next funding period we plan to proceed with the corresponding calculation for the ``nondiagonal'' currents. The results will be useful in connection with lattice gauge theory (similarly to our analysis of the diagona current) and other applications.

A completely different research topic within project A1 is concerned with
electroweak interactions in the TeV region, specifically the appearance
of ``Sudakov logarithms''. In this energy region exclusive cross
sections are modified by large, double-logarithmic corrections
∝α_{ W}ln^{2} (s/M^{2}_{w}).
During the past four years the methods,
developed and applied to four-fermion processes earlier by our group,
have been extended to gauge boson production.
During the next four years the emphasis will shift to gauge boson
production relevant for the LHC.

Current techniques for the algebraic simplification and evaluation of Feynman integrals are based on recursion algorithms obtained by using Integration-by-Parts identities to relate the millions of individual integrals to a small set of basic ``master'' integrals. The development of new and refined program packages, using both well tested and new algorithms is therefore an essential part of project A1, closely connected to and partly also described in project A2.

Last Change: 14th June 2011